Intereting Posts

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Differential equations and Fourier and Laplace transforms
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Find limit without using L'hopital or Taylor's series
Foundation for analysis without axiom of choice?
The king comes from a family of 2 children. What is the probability that the other child is his sister?
A possible inequality related to binomial theorem (or, convex/concave functions)
Accidents of small $n$
Gradient of a vector field?
Number of distinct prime factors, omega(n)
Interesting “real life” applications of serious theorems
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.

Is the following statement is true?

There are at least three mutually non-isomorphic rings with $4$ elements.

I have no idea or counterexample at the moment. Please help. So far I know about that a group of order $4$ is abelian and there are two non isomorphic groups of order $4$ like $K_4(non cyclic)$ and $\mathbb Z_4(cyclic)$.

- Prove that if $i$ and $j$ are integers, the $\langle a^i \rangle = \langle a^j \rangle$ iff $i=\pm j$
- Non-commutative rings without identity
- $\mathbb{Q}/\mathbb{Z}$ is divisible
- Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
- Groups of units of $\mathbb{Z}\left$
- What is it to be normal?

- Irreducibility of $x^p-x-c$
- Show that $\mathbb{Q}(\sqrt{2}+\sqrt{5})=\mathbb{Q}(\sqrt{2},\sqrt{5})$
- Unique factorization domain that is not a Principal ideal domain
- Zero-divisors and units in $\mathbb Z_4$
- Transcendental Extensions. $F(\alpha)$ isomorphic to $F(x)$
- Extending Herstein's Challenging Exercise to Modules
- Why teach linear algebra before abstract algebra?
- Show that $G$ is cyclic
- Examples for almost-semirings without absorbing zero
- Integral domain without unity has prime characteristic?

By ring, I always mean unital ring. Each of the following rings has four elements:

$R_1 = \mathbb{Z}/4~, ~R_2 = \mathbb{F}_2 \times \mathbb{F}_2 = \mathbb{F}_2[x]/(x^2+x)~, ~R_3 = \mathbb{F}_4=\mathbb{F}_2[x]/(x^2+x+1)~, ~R_4 = \mathbb{F}_2[x]/(x^2)$

They are non-isomorphic because only $R_3$ is a field, only $R_1$ has characteristic $\neq 2$, and only $R_2,R_3$ are reduced.

Conversely, let $R$ be a ring with four elements. If $a \in R \setminus \{0,1\}$, then the centralizer of $a$ is a subgroup of $(R,+)$ with at least three elements $0,1,a$, so by Lagrange also the fourth element has to commute with $a$. Thus, $R$ is commutative. If $R$ is reduced, then it is a finite product of local artinian reduced rings, i.e. fields, so that $R \cong R_2$ or $R \cong R_3$. If $R$ is not reduced, there is some $a \in R \setminus \{0\}$ such that $a^2=0$. Since $0,1,a,a+1$ are pairwise distinct, these are the elements of $R$. If $2=0$, then we get an injective homomorphism $\mathbb{F}_2[x]/(x^2) \to R, x \mapsto a$. Since both sides have four elements, it is an isomorphism. If $2 \neq 0$, the characteristic has to be $4$, i.e. we get an embedding $\mathbb{Z}/4 \to R$, which again has to be an isomorphism.

Of course, this classification can also be obtained by more elementary methods. For other orders, see:

- http://oeis.org/A037291: Number of rings with $n$ elements
- http://oeis.org/A127707: Number of commutative rings with $n$ elements
- http://oeis.org/A127708: Number of non-commutative rings with $n$ elements
- http://oeis.org/A027623: Number of rngs with $n$ elements
- http://oeis.org/A037289: Number of commutative rngs with $n$ elements

The smallest non-commutative ring has $8$ elements and is given by $\begin{pmatrix} \mathbb{F}_2 & \mathbb{F}_2 \\ 0 & \mathbb{F}_2 \end{pmatrix} \subseteq M_2(\mathbb{F}_2)$.

Consider the ring $R = \Bbb{Z}/2\Bbb{Z}[i]$. Alternatively $R$ can be constructed as a quotient

$$R \cong \Bbb{Z}[x]/(2,x^2+1).$$

As a ring $R$ is not isomorphic to either $S = \Bbb{Z}/2\Bbb{Z} \times \Bbb{Z}/2\Bbb{Z}$ or $ T = \Bbb{Z}/4\Bbb{Z}$.

**Edit:** Perhaps I should add why the ring $R$ is not isomorphic to either $S$ or $T$. Firstly by counting orders of elements $R$ cannot be isomorphic to $T$; $T$ has an element of order 4 while $R$ does not. So now the penultimate question is why is $R$ is not isomorphic to $S$? As groups they are certainly isomorphic but as rings they can’t be. The reason is because the presence of $i$ means that the multiplication in $\Bbb{Z}/2\Bbb{Z}[i]$ is not the same as the multiplication in $S$ which is the usual one coming from the product ring structure.

In view of this we see that $R$ has non-trivial nilpotent elements, $(1+i)^2 = 1 – 2i +i^2 = 1 – 1 = 0$ while obviously $\text{nilrad}$ $S = 0$. Thus $R \not\cong S$.

These can be exhaustively enumerated using alg. To enumerate the number of non-isomorphic rings of orders $1,2,\ldots,8$, we enter:

```
./alg theories/ring.th --size 1-8 --count
```

which outputs:

```
# Theory ring
Constant 0.
Unary ~.
Binary + *.
Axiom plus_commutative: x + y = y + x.
Axiom plus_associative: (x + y) + z = x + (y + z).
Axiom zero_neutral_left: 0 + x = x.
Axiom zero_neutral_right: x + 0 = x.
Axiom negative_inverse: x + ~ x = 0.
Axiom negative_inverse: ~ x + x = 0.
Axiom zero_inverse: ~ 0 = 0.
Axiom inverse_involution: ~ (~ x) = x.
Axiom mult_associative: (x * y) * z = x * (y * z).
Axiom distrutivity_right: (x + y) * z = x * z + y * z.
Axiom distributivity_left: x * (y + z) = x * y + x * z.
size | count
-----|------
1 | 1
2 | 2
3 | 2
4 | 11
5 | 2
6 | 4
7 | 2
8 | 52
Check the numbers [2, 2, 11, 2, 4, 2, 52](http://oeis.org/search?q=2,2,11,2,4,2,52) on-line at oeis.org
```

If we want it to print out the addition and multiplication tables, we can remove `--count`

from the command line:

```
./alg theories/ring.th --size 4
```

If we want to work with unital rings, we can use:

```
./alg theories/unital_ring.th --size 1-8 --count
```

which gives:

```
# Theory unital_ring
Theory unital_ring.
Constant 0 1.
Unary ~.
Binary + *.
Axiom plus_commutative: x + y = y + x.
Axiom plus_associative: (x + y) + z = x + (y + z).
Axiom zero_neutral_left: 0 + x = x.
Axiom negative_inverse: x + ~ x = 0.
Axiom mult_associative: (x * y) * z = x * (y * z).
Axiom one_unit_left: 1 * x = x.
Axiom one_unit_right: x * 1 = x.
Axiom distrutivity_right: (x + y) * z = x * z + y * z.
Axiom distributivity_left: x * (y + z) = x * y + x * z.
# Consequences of axioms that make alg run faster:
Axiom zero_neutral_right: x + 0 = x.
Axiom negative_inverse: ~ x + x = 0.
Axiom zero_inverse: ~ 0 = 0.
Axiom inverse_involution: ~ (~ x) = x.
Axiom mult_zero_left: 0 * x = 0.
Axiom mult_zero_right: x * 0 = 0.
size | count
-----|------
1 | 0
2 | 1
3 | 1
4 | 4
5 | 1
6 | 1
7 | 1
8 | 11
Check the numbers [1, 1, 4, 1, 1, 1, 11](http://oeis.org/search?q=1,1,4,1,1,1,11) on-line at oeis.org
```

- Weak-to-weak continuous operator which is not norm-continuous
- On the number of ways of writing an integer as a sum of 3 squares using triangular numbers.
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